Here we use the notation \(A \lesssim B\)  if there is a universal constant \(C>0\)  so that \(A \le C \cdot B\) .
The statement differs in several aspects to the original statement of [1]}.
We discuss and justify the changes in Appendix .
Intuitively, \(\pazocal {D}(\mathbf {A},\mathbf {x}_0)\)  is simply the outcome of a Brownian motion starting at \(\mathbf {x}_0\)  that freezes coordinates as soon as they hit \(+1\)  or \(-1\)  and it freezes constraints if \(|\left<\mathbf {A}_i,\mathbf {x}-\mathbf {x}_0\right>| = \Delta \) . For example, if \(\mathbf {A} \in \lbrace  0,1\rbrace ^{m \times n}\)  is the
incidence matrix in Spencer's setting, then one may choose \(\Delta := \Theta (\sqrt{n\log (\frac{2m}{n})})\)  and obtain a partial coloring of discrepancy \(\Delta \)  that colors at least half the coordinatesOne can then obtain a full coloring by iterating the argument \(O(\log n)\)  times. In the \(i\) th such iteration (with \(i \ge 0\) ) there are at most \(n/2^i\)  elements uncolored, so the suffered discrepancy decreases to \(O(\sqrt{ (n/2^{i}) \log \frac{2m}{n/2^i} })\) . Summing over these terms gives a convergent sum of value \(O(\Delta )\) ..
